Painlev´ e equations Geodesic lengths Decorated character variety Quantisation PVI as isomonodromic deformation-I Painlev´ e sixth equation The Painlev´ e VI equation describes the isomonodromic deformations of the rank 2 meromorphic connections on P 1 with simple poles. d Y ( A 1 ( z ) + A 2 ( z ) λ − t + A 3 ( z ) ) d λ = λ − 1 , Y , λ ∈ C \ { 0 , t , 1 } (1) λ where A 1 , A 2 , A 3 ∈ sl 2 ( C ) , A 1 + A 2 + A 3 = − A ∞ , diagonal. Fundamental matrix: Y ∞ ( λ ) = (1 + O ( 1 λ )) λ A ∞ . Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation PVI as isomonodromic deformation-I Painlev´ e sixth equation The Painlev´ e VI equation describes the isomonodromic deformations of the rank 2 meromorphic connections on P 1 with simple poles. d Y ( A 1 ( z ) + A 2 ( z ) λ − t + A 3 ( z ) ) d λ = λ − 1 , Y , λ ∈ C \ { 0 , t , 1 } (1) λ where A 1 , A 2 , A 3 ∈ sl 2 ( C ) , A 1 + A 2 + A 3 = − A ∞ , diagonal. Fundamental matrix: Y ∞ ( λ ) = (1 + O ( 1 λ )) λ A ∞ . Monodromy matrices γ j ( Y ∞ ) = Y ∞ M j Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation PVI as isomonodromic deformation-I Painlev´ e sixth equation The Painlev´ e VI equation describes the isomonodromic deformations of the rank 2 meromorphic connections on P 1 with simple poles. d Y ( A 1 ( z ) + A 2 ( z ) λ − t + A 3 ( z ) ) d λ = λ − 1 , Y , λ ∈ C \ { 0 , t , 1 } (1) λ where A 1 , A 2 , A 3 ∈ sl 2 ( C ) , A 1 + A 2 + A 3 = − A ∞ , diagonal. Fundamental matrix: Y ∞ ( λ ) = (1 + O ( 1 λ )) λ A ∞ . Monodromy matrices γ j ( Y ∞ ) = Y ∞ M j Describes by generators of the fundamental group under the anti-isomorphism P 1 \{ 0 , t , 1 , ∞} , λ 1 ( ) ρ : π 1 → SL 2 ( C ) . Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation PVI as isomonodromic deformation-II eigen( M j ) = eigen(exp(2 π iA j ) We fix the base point λ 1 at infinity and the generators of the fundamental group to be γ 1 , γ 2 , γ 3 such that γ j encircles only the pole i once and are oriented in such a way that M 1 M 2 M 3 M ∞ = I , M ∞ = exp(2 π iA ∞ ) . (2) Eigenvalues of A j are ( θ j , − θ j ) , j = 0 , t , 1 , ∞ . α := ( θ ∞ − 1 / 2) 2 ; β := − θ 2 0 ; γ := θ 2 δ := (1 / 4 − θ t ) 2 . 1 ; Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation PVI as isomonodromic deformation-III Let: G j := Tr ( M j ) = 2 cos( πθ j ) , j = 0 , t , 1 , ∞ , The Riemann-Hilbert correspondence F ( θ 0 , θ t , θ 1 , θ ∞ ) / G → M ( G 1 , G 2 , G 3 , G ∞ ) / SL 2 ( C ) , where G is the gauge group, is defined by associating to each Fuchsian system its monodromy representation class. The representation space M ( G 1 , G 2 , G 3 , G ∞ ) is realised as an affine cubic surface ( Jimbo) x 1 x 2 x 3 + x 2 1 + x 2 2 + x 2 3 + ω 1 x 1 + ω 2 x 2 + ω 3 x 3 + ω 4 = 0 , (3) where: Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation PVI as isomonodromic deformation-IV x 1 = Tr ( M 2 M 3 ) , x 2 = Tr ( M 1 M 3 ) , x 3 = Tr ( M 1 M 2 ) . and − ω i := G k G j + G i G ∞ , i ̸ = k , j , ω ∞ = G 2 1 + G 2 2 + G 2 3 + G 2 ∞ + G 1 G 2 G 3 G ∞ − 4 . Iwasaki proved that the triple ( x 1 , x 2 , x 3 ) satisfying the cubic relation (3) provides a set of coordinates on a large open subset S ⊂ M ( G 1 , G 2 , G 3 , G ∞ ) . In what follows, we restrict to such open set. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation General Affine Cubic The main object studied in this talk is the affine irreducible cubic surface M φ := Spec ( C [ x 1 , x 2 , x 3 ] / ⟨ φ = 0 ⟩ ) where φ = x 1 x 2 x 3 + ϵ ( d ) 1 + ϵ ( d ) 2 + ϵ ( d ) 3 + ω ( d ) 1 x 1 + ω ( d ) 2 x 2 + ω ( d ) 3 x 3 + ω ( d ) 1 x 2 2 x 2 3 x 2 = 0 , 4 (4) According to Saito and Van der Put , all monodromy manifolds M ( d ) have the form of M φ for φ form the list above. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Affine Cubic as it is -1: In singularity theory - the universal unfolding of the D 4 singularity. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Affine Cubic as it is -1: In singularity theory - the universal unfolding of the D 4 singularity. Oblomkov : the quantisation of the D 4 affine cubic surface M φ coincides with spherical subalgebra of the generalised rank 1 double affine Hecke algebra H (or Cherednick algebra of type C 1 C ν 1 ) Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Affine Cubic as it is -1: In singularity theory - the universal unfolding of the D 4 singularity. Oblomkov : the quantisation of the D 4 affine cubic surface M φ coincides with spherical subalgebra of the generalised rank 1 double affine Hecke algebra H (or Cherednick algebra of type C 1 C ν 1 ) In algebraic geometry - projective completion: φ := { ( u , v , w , t ) ∈ P 3 | x 2 1 t + x 2 2 t + x 2 M � 3 t − x 1 x 2 x 3 + + ω 3 x 1 t 2 + ω 2 x 2 t 2 + ω 3 x 3 t 2 + ω 4 t 3 = 0 } is a del Pezzo surface of degree three and differs from it by three smooth lines at infinity forming a triangle [ Oblomkov ] t = 0 , x 1 x 2 x 3 = 0 . Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation This family of cubics is a variety ω ) ∈ C 3 × Ω) : φ (¯ M φ = { (¯ x , ¯ x , ¯ ω ) = 0 } where x = ( x 1 , x 2 , x 3 ) , ¯ ω = ( ω 1 , ω 2 , ω 3 , ω 4 ) and the ”¯ ¯ x − forgetful” projection π : M φ → Ω : π (¯ x , ¯ ω ) = ¯ ω. This projection defines a family of affine cubics with generically non–singular fibres π − 1 (¯ ω ) The cubic surface M φ has a volume form ϑ ¯ ω given by the Poincar´ e residue formulae: dx 1 ∧ dx 2 dx 2 ∧ dx 3 dx 3 ∧ dx 1 ϑ ¯ ω = ω ) / ( ∂ x 3 ) = ω ) / ( ∂ x 1 ) = ω ) / ( ∂ x 2 ) . (5) ( ∂φ ¯ ( ∂φ ¯ ( ∂φ ¯ Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation The volume form is a holomorphic 2-form on the non-singular part of M φ and it has singularities along the singular locus. This form defines the Poisson brackets on the surface in the usual way as ω = ∂φ ¯ ω { x 1 , x 2 } ¯ (6) ∂ x 3 The other brackets are defined by circular transposition of x 1 , x 2 , x 3 . For ( i , j , k ) = (1 , 2 , 3): ω = ∂φ ¯ ω = x i x j + 2 ϵ d i x k + ω d { x i , x j } ¯ (7) i ∂ x k and the volume form (5) reads as dx i ∧ dx j dx i ∧ dx j ϑ ¯ ω = ω /∂ x k ) = i ) . (8) ( x i x j + 2 ϵ d i x k + ω d ( ∂φ ¯ Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Observe that for any φ ∈ C [ x 1 , x 2 , x 3 ] the following formulae define a Poisson bracket on C [ x 1 , x 2 , x 3 ]: ∂φ { x i , x i +1 } = , x i +3 = x i , (9) ∂ x i +2 and φ itself is a central element for this bracket, so that the quotient space M φ := Spec ( C [ x 1 , x 2 , x 3 ] / ⟨ φ =0 ⟩ ) inherits the Poisson algebra structure [ Nambu ∼ 70]. Today we are going to re-parametrize and to quantize it. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Affine Cubic as it is -2: In the Painlev´ e context (the dynamic on the family of PVI monodromy surfaces) this cubic was considered by S. Cantat et F. Loray and by M. Inaba, K. Iwasaki and M.Saito. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Affine Cubic as it is -2: In the Painlev´ e context (the dynamic on the family of PVI monodromy surfaces) this cubic was considered by S. Cantat et F. Loray and by M. Inaba, K. Iwasaki and M.Saito. PVI ( ˜ D 4 ) cubic with only ω 4 ̸ = 0 admits the log-canonical symplectic structure ¯ ϑ = du ∧ dv under isomorphism uv C ∗ × C ∗ /ı → M φ by ( u , v ) → ( x 1 = − ( u + 1 u ) , x 2 = − ( v + 1 v ) , x 3 = − ( uv + 1 uv )) and ı : C ∗ → C ∗ is the involution ı ( u ) = 1 u , ı ( v ) = 1 v . Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Affine Cubic as it is -2: In the Painlev´ e context (the dynamic on the family of PVI monodromy surfaces) this cubic was considered by S. Cantat et F. Loray and by M. Inaba, K. Iwasaki and M.Saito. PVI ( ˜ D 4 ) cubic with only ω 4 ̸ = 0 admits the log-canonical symplectic structure ¯ ϑ = du ∧ dv under isomorphism uv C ∗ × C ∗ /ı → M φ by ( u , v ) → ( x 1 = − ( u + 1 u ) , x 2 = − ( v + 1 v ) , x 3 = − ( uv + 1 uv )) and ı : C ∗ → C ∗ is the involution ı ( u ) = 1 u , ı ( v ) = 1 v . The family (3) can be ”uniformize” by some analogues of theta-functions related to toric mirror data on log-Calabi-Yau surfaces ( M. Gross, P. Hacking and S.Keel (see Example 5.12 of ”Mirror symmetry for log-Calabi-Yau varieties I, arXiv:1106.4977). Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation The other Painlev´ e equations The PVI monodromy manifold is the SL 2 ( C )–character variety of a four holed Riemann sphere. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation The other Painlev´ e equations The PVI monodromy manifold is the SL 2 ( C )–character variety of a four holed Riemann sphere. What are the underlying Riemann surfaces for the other Painlev´ e equations? Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation The other Painlev´ e equations The PVI monodromy manifold is the SL 2 ( C )–character variety of a four holed Riemann sphere. What are the underlying Riemann surfaces for the other Painlev´ e equations? Is there a ”toric” (or cluster algebra) structure on it? Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
� � � � � � � � � Painlev´ e equations Geodesic lengths Decorated character variety Quantisation The other Painlev´ e equations The PVI monodromy manifold is the SL 2 ( C )–character variety of a four holed Riemann sphere. What are the underlying Riemann surfaces for the other Painlev´ e equations? Is there a ”toric” (or cluster algebra) structure on it? Use the confluence scheme of the Painlev´ e equations. � P D 7 � P D 8 P D 6 III III III � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � P V I � P V P deg P JM � P I V II � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � P IV � P F N II Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
� � �� � �� � � �� � � � � � � � � �� � � � � � � � � � � � Painlev´ e equations Geodesic lengths Decorated character variety Quantisation ”Confluented ”Poisson algebras Poiss D 6 Poiss D 7 Poiss D 8 III � � III III � � � Poiss V Poiss deg Poiss JM Poiss VI � Poiss I � � V II � � Poiss FN Poiss IV II Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Poincar´ e-Katz invariants and Stokes rays Painlev´ e eqs no. of cusps Katz invariants no. Stokes rays pole–orders for φ PVI (0 , 0 , 0 , 0) (0 , 0 , 0 , 0) (0 , 0 , 0 , 0) (2 , 2 , 2 , 2) PV (0 , 0 , 2) (0 , 0 , 1) (0 , 0 , 2) (2 , 2 , 4) PV deg (0 , 0 , 1) (0 , 0 , 1 / 2) (0 , 0 , 1) (2 , 2 , 3) PIV (0 , 4) (0 , 2) (0 , 4) (2 , 6) PIII D 6 (0 , 2 , 2) (0 , 1 , 1) (0 , 2 , 2) (2 , 4 , 4) PIII D 7 (0 , 1 , 2) (0 , 1 / 2 , 1) (0 , 1 , 2) (2 , 3 , 4) PIII D 8 (0 , 1 , 1) (0 , 1 / 2 , 1 / 2) (0 , 1 , 1) (2 , 3 , 3) PII FN (0 , 3) (0 , 3 / 2) (0 , 3) (2 , 5) PII MJ 6 3 6 8 PI 5 5 / 2 5 7 Table: For each Painlev´ e isomonodromic problem, this table displays the number of cusps on each hole for the corresponding Riemann surface, the Katz invariants associated to the corresponding flat connection, the number of Stokes rays in the linear system defined by the flat connection and the number of poles of the quadratic differential φ defined by the linear system. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Diagonalizable (non-ramified) case Notation: the fundamental matrix at an irregular singular point λ k has the form ( e Q k ( λ ) ) 0 Y k = G k ( λ ) λ Λ k e − Q k ( λ ) 0 A ( λ ) Casimirs extended exponents dim ( C ) A 0 A 1 Q ∞ = t PV λ + λ − 1 + A ∞ eigen( A 0 ),eigen( A 1 ),Λ ∞ 2 λ 7 Q ∞ = λ 2 + t A 0 PIV λ + A 1 + A ∞ λ eigen( A 0 ),Λ ∞ 2 λ 8 PIII D 6 A 0 λ 2 + A 1 Q ∞ = t 2 λ, Q 0 = t 1 λ + A ∞ Λ 0 , Λ ∞ 8 2 λ Q ∞ = λ 3 + t PII MJ A 0 + A 1 λ + A ∞ λ 2 Λ ∞ 2 λ 9 Table: Here Q k ( λ ) is polynomial in ( λ − λ k ) of order n − 1 with n being the order of λ k and Λ k is the formal monodromy (diagonal). Expand A ( λ ) near λ k to calculate Q k ( λ ) and Λ k , then diagonalize it using the gauge transformation G k ( λ ) . . Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Non-diagonalizable (ramified) case The same notations. A ( λ ) Casimirs extended exponents dim ( C ) √ A 0 A 1 Q ∞ = t PV deg λ + λ − 1 + A ∞ eigen( A 0 ),eigen( A 1 ),Λ ∞ λ 5 2 PIII D 7 A 0 λ 2 + A 1 1 λ , Q ∞ = t λ + A ∞ Λ 0 , Λ ∞ Q 0 = √ 2 λ 6 √ PIII D 8 A 0 λ 2 + A 1 1 λ + A ∞ Λ 0 , Λ ∞ Q ∞ = λ, Q 0 = 4 √ √ λ Q ∞ = λ 3 / 2 + t PII FN A 0 λ + A 1 + A ∞ λ eigen( A 0 ), Λ ∞ λ 6 √ 2 Q ∞ = λ 5 / 2 + t A 0 + A 1 λ + A ∞ λ 2 PI Λ ∞ λ 7 2 Table: Here dim PIII D 7 = dim PIII − 2 , dim PIII D 8 = dim PV deg = dim PV − 2 , dim PII FN = dim PIV − 2 , dim PI = dim PII JM − 2 . dim PIII D 7 − 2 , Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Geometric confluence scheme of the Painlev´ e equations: application. ( P. Gavrylenko, O. Lisovyy, arXiv:1608.00958) V deg III(D ) 8 V III(D ) 7 VI III(D ) 6 II FN IV I II JM Figure 3: CMR confluence diagram for Painlevé equations. Gauss Whittaker Bessel Figure 4: Some solvable RHPs in rank N = 2: Gauss hypergeometric (3 regular punc- tures), Whittaker (1 regular + 1 of Poincaré rank 1) and Bessel (1 regular + 1 of rank 1 2 ). Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Basic ideas:history The character variety of a Riemann sphere with 4 holes Hom ( π 1 ( P 1 \ { 0 , t , 1 , ∞} ); SL 2 ( C )) / SL 2 ( C ) is the monodromy cubic of the Painlev´ e VI ( Goldman-Toledo). Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Basic ideas:history The character variety of a Riemann sphere with 4 holes Hom ( π 1 ( P 1 \ { 0 , t , 1 , ∞} ); SL 2 ( C )) / SL 2 ( C ) is the monodromy cubic of the Painlev´ e VI ( Goldman-Toledo). The ”wild” character varieties, fissions and meromorphic connections ( Ph. Boalch 2011-2015). Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Basic ideas:history The character variety of a Riemann sphere with 4 holes Hom ( π 1 ( P 1 \ { 0 , t , 1 , ∞} ); SL 2 ( C )) / SL 2 ( C ) is the monodromy cubic of the Painlev´ e VI ( Goldman-Toledo). The ”wild” character varieties, fissions and meromorphic connections ( Ph. Boalch 2011-2015). Quasi-Poisson structures ( A. Alexeev, Y. Kosmann-Schwarzbach, E.Meinrenken, M. Van den Bergh 1999 -2005). Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Basic ideas:history The character variety of a Riemann sphere with 4 holes Hom ( π 1 ( P 1 \ { 0 , t , 1 , ∞} ); SL 2 ( C )) / SL 2 ( C ) is the monodromy cubic of the Painlev´ e VI ( Goldman-Toledo). The ”wild” character varieties, fissions and meromorphic connections ( Ph. Boalch 2011-2015). Quasi-Poisson structures ( A. Alexeev, Y. Kosmann-Schwarzbach, E.Meinrenken, M. Van den Bergh 1999 -2005). Moduli spaces for quilted surfaces. ( D. Li-Bland, P. Severa 2013). Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Basic ideas:history The character variety of a Riemann sphere with 4 holes Hom ( π 1 ( P 1 \ { 0 , t , 1 , ∞} ); SL 2 ( C )) / SL 2 ( C ) is the monodromy cubic of the Painlev´ e VI ( Goldman-Toledo). The ”wild” character varieties, fissions and meromorphic connections ( Ph. Boalch 2011-2015). Quasi-Poisson structures ( A. Alexeev, Y. Kosmann-Schwarzbach, E.Meinrenken, M. Van den Bergh 1999 -2005). Moduli spaces for quilted surfaces. ( D. Li-Bland, P. Severa 2013). Stokes groupoides ( M. Gualtieri, S. Li, B. Pym 2014). Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Basic ideas: Painlev´ e context Poisson structures and Isomonodromic deformations on the sphere and torus ( Korotkin-Samtleben, Chekhov-Mazzocco 1995, 2010). Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Basic ideas: Painlev´ e context Poisson structures and Isomonodromic deformations on the sphere and torus ( Korotkin-Samtleben, Chekhov-Mazzocco 1995, 2010). Wild character variety for PV ( J.P. Ramis, E. Paul. 2015). Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Basic ideas: Painlev´ e context Poisson structures and Isomonodromic deformations on the sphere and torus ( Korotkin-Samtleben, Chekhov-Mazzocco 1995, 2010). Wild character variety for PV ( J.P. Ramis, E. Paul. 2015). The confluent Painlev´ e monodromy manifolds are ”decorated character varieties” ( Chekhov-Mazzocco -R.2015). Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Basic ideas: Painlev´ e context Poisson structures and Isomonodromic deformations on the sphere and torus ( Korotkin-Samtleben, Chekhov-Mazzocco 1995, 2010). Wild character variety for PV ( J.P. Ramis, E. Paul. 2015). The confluent Painlev´ e monodromy manifolds are ”decorated character varieties” ( Chekhov-Mazzocco -R.2015). The real slice of the SL 2 ( C ) character variety is the Teichm¨ uller space. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Basic ideas: Painlev´ e context Poisson structures and Isomonodromic deformations on the sphere and torus ( Korotkin-Samtleben, Chekhov-Mazzocco 1995, 2010). Wild character variety for PV ( J.P. Ramis, E. Paul. 2015). The confluent Painlev´ e monodromy manifolds are ”decorated character varieties” ( Chekhov-Mazzocco -R.2015). The real slice of the SL 2 ( C ) character variety is the Teichm¨ uller space. The shear coordinates on the Teichm¨ uller space can be complexified ⇒ coordinate description for the character variety. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Basic ideas: Painlev´ e context Poisson structures and Isomonodromic deformations on the sphere and torus ( Korotkin-Samtleben, Chekhov-Mazzocco 1995, 2010). Wild character variety for PV ( J.P. Ramis, E. Paul. 2015). The confluent Painlev´ e monodromy manifolds are ”decorated character varieties” ( Chekhov-Mazzocco -R.2015). The real slice of the SL 2 ( C ) character variety is the Teichm¨ uller space. The shear coordinates on the Teichm¨ uller space can be complexified ⇒ coordinate description for the character variety. To visualize the confluence and the ”decoration” we shall introduce two moves correspond to certain asymptotics in the (complexified) shear coordinates. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Two mouves Hooking holes: Pinching two sides of the same hole: Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Figure: The process of confluence of two holes on the Riemann sphere with four holes. Chewing-gum move: hook two holes together and stretch to infinity by keeping the area between them finite (see Fig.). As a result we obtain a Riemann sphere with one less hole, but with two new cusps on the boundary of this hole. The red geodesic line which was initially closed becomes infinite, therefore two horocycles (the green dashed circles) must be introduced in order to measure its length. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Teichm¨ uller space For Riemann surfaces with holes: Hom ( π 1 (Σ) , P SL 2 ( R )) / GL 2 ( R ) . Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Teichm¨ uller space For Riemann surfaces with holes: Hom ( π 1 (Σ) , P SL 2 ( R )) / GL 2 ( R ) . Idea: Teichm¨ uller theory for a Riemann surfaces with holes is well understood. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Teichm¨ uller space For Riemann surfaces with holes: Hom ( π 1 (Σ) , P SL 2 ( R )) / GL 2 ( R ) . Idea: Teichm¨ uller theory for a Riemann surfaces with holes is well understood. Take confluences of holes to obtain cusps. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Teichm¨ uller space For Riemann surfaces with holes: Hom ( π 1 (Σ) , P SL 2 ( R )) / GL 2 ( R ) . Idea: Teichm¨ uller theory for a Riemann surfaces with holes is well understood. Take confluences of holes to obtain cusps. Develop bordered cusped Teichm¨ uller theory asymptotically. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Teichm¨ uller space For Riemann surfaces with holes: Hom ( π 1 (Σ) , P SL 2 ( R )) / GL 2 ( R ) . Idea: Teichm¨ uller theory for a Riemann surfaces with holes is well understood. Take confluences of holes to obtain cusps. Develop bordered cusped Teichm¨ uller theory asymptotically. This will provide a cluster algebra of geometric type Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Coordinates: geodesic lengths Theorem The geodesic length functions (G γ := Tr γ = 2 cosh ( l γ ) ) form an algebra with multiplication: G γ G ˜ γ = G γ ˜ γ + G γ ˜ γ − 1 . ( Tr ( AB ) + Tr ( AB − 1 ) = Tr ( A ) Tr ( B ) ∀ A , B ∈ SL 2 ( C )) Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Coordinates: geodesic lengths Theorem The geodesic length functions (G γ := Tr γ = 2 cosh ( l γ ) ) form an algebra with multiplication: G γ G ˜ γ = G γ ˜ γ + G γ ˜ γ − 1 . ( Tr ( AB ) + Tr ( AB − 1 ) = Tr ( A ) Tr ( B ) ∀ A , B ∈ SL 2 ( C )) ˜ γ γ Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Coordinates: geodesic lengths Theorem The geodesic length functions (G γ := Tr γ = 2 cosh ( l γ ) ) form an algebra with multiplication: G γ G ˜ γ = G γ ˜ γ + G γ ˜ γ − 1 . ( Tr ( AB ) + Tr ( AB − 1 ) = Tr ( A ) Tr ( B ) ∀ A , B ∈ SL 2 ( C )) = ˜ γ γ Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Coordinates: geodesic lengths Theorem The geodesic length functions (G γ := Tr γ = 2 cosh ( l γ ) ) form an algebra with multiplication: G γ G ˜ γ = G γ ˜ γ + G γ ˜ γ − 1 . ( Tr ( AB ) + Tr ( AB − 1 ) = Tr ( A ) Tr ( B ) ∀ A , B ∈ SL 2 ( C )) = + ˜ γ γ − 1 γ ˜ γ γ ˜ γ Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Poisson structure γ } = 1 γ − 1 { G γ , G ˜ 2 G γ ˜ 2 G γ ˜ γ − 1 . { } = 1 − 1 2 2 γ ˜ γ − 1 ˜ γ ˜ γ γ γ Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Ptolemy Relation-1 ˜ γ a G ˜ γ e G ˜ γ f = G ˜ γ a G ˜ γ c + G ˜ γ b G ˜ ˜ γ f ˜ γ e γ d γ b ˜ γ d ˜ γ c ˜ Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Ptolemy Relation-2 aa ′ + bb ′ = cc ′ a c ′ c b ′ b a ′ Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Ptolemy Relation ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) → ( x ′ 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) x 1 x ′ 1 = x 9 x 7 + x 8 x 2 • x 9 x ′ • x 1 1 x 8 • x 2 x 4 x 7 • x 3 x 5 • • x 6 Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Ptolemy Relation ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) → ( x ′ 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) x 1 x ′ 1 = x 9 x 7 + x 8 x 2 • x 9 x ′ • x 1 1 x 8 • x 2 x 4 x 7 • x 3 x 5 • • x 6 Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Ptolemy Relation ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) → ( x ′ 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) x 1 x ′ 1 = x 9 x 7 + x 8 x 2 • x 9 x ′ • x 1 1 x 8 • x 2 x 4 x 7 • x 3 x 5 • • x 6 Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Ptolemy Relation ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) → ( x ′ 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) x 1 x ′ 1 = x 9 x 7 + x 8 x 2 • x 9 x ′ • x 1 1 x 8 • x 2 x 4 x 7 • x 3 x 5 • • x 6 Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Ptolemy Relation ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) → ( x ′ 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) x 1 x ′ 1 = x 9 x 7 + x 8 x 2 • x 9 x ′ • 1 x 8 • x 2 x 4 x 7 • x 3 x 5 • • x 6 Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Ptolemy Relation ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) → ( x ′ 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) ( x ′ 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) → ( x ′ 1 , x 2 , x ′ 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) • x 9 x ′ • 1 x 8 • x 2 x 4 x 7 • x 3 x 5 • • x 6 Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Ptolemy Relation ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) → ( x ′ 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) ( x ′ 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) → ( x ′ 1 , x ′ 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) • x 9 x ′ • 1 x 8 • x ′ x 2 2 x 4 x 7 • x 3 x 5 • • x 6 Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Cluster algebra We call a set of n numbers ( x 1 , . . . , x n ) a cluster of rank n . Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Cluster algebra We call a set of n numbers ( x 1 , . . . , x n ) a cluster of rank n . A seed consists of a cluster and an exchange matrix B , i.e. a skew–symmetrisable matrix with integer entries. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Cluster algebra We call a set of n numbers ( x 1 , . . . , x n ) a cluster of rank n . A seed consists of a cluster and an exchange matrix B , i.e. a skew–symmetrisable matrix with integer entries. A mutation is a transformation n ), µ i : B → B ′ where µ i : ( x 1 , x 2 , . . . , x n ) → ( x ′ 1 , x ′ 2 , . . . , x ′ x b ij x − b ij x i x ′ ∏ ∏ x ′ i = + , j = x j ∀ j ̸ = i . j j j : b ij > 0 j : b ij < 0 Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Cluster algebra We call a set of n numbers ( x 1 , . . . , x n ) a cluster of rank n . A seed consists of a cluster and an exchange matrix B , i.e. a skew–symmetrisable matrix with integer entries. A mutation is a transformation n ), µ i : B → B ′ where µ i : ( x 1 , x 2 , . . . , x n ) → ( x ′ 1 , x ′ 2 , . . . , x ′ x b ij x − b ij x i x ′ ∏ ∏ x ′ i = + , j = x j ∀ j ̸ = i . j j j : b ij > 0 j : b ij < 0 Definition A cluster algebra of rank n is a set of all seeds ( x 1 , . . . , x n , B ) related to one another by sequences of mutations µ 1 , . . . , µ k . The cluster variables x 1 , . . . , x k are called exchangeable, while x k +1 , . . . , x n are called frozen. [Fomin-Zelevnsky 2002]. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Example Cluster algebra of rank 9 with 3 exchangeable variables x 1 , x 2 , x 3 and 6 frozen ones x 4 , . . . , x 9 . ( x 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) → ( x ′ 1 , x 2 , x 3 , x 4 , x 5 , x 6 , x 7 , x 8 , x 9 ) x 1 x ′ 1 = x 9 x 7 + x 8 x 2 • x 9 x ′ • x 1 1 x 8 • x 2 x 4 x 7 • x 3 x 5 • • x 6 Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Outline Are all cluster algebras of geometric origin? Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Outline Are all cluster algebras of geometric origin? Introduce bordered cusps Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Outline Are all cluster algebras of geometric origin? Introduce bordered cusps Geodesics length functions on a Riemann surface with bordered cusps form a cluster algebra. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Outline Are all cluster algebras of geometric origin? Introduce bordered cusps Geodesics length functions on a Riemann surface with bordered cusps form a cluster algebra. All Berenstein-Zelevinsky cluster algebras are geometric Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Poisson bracket Introduce cusped laminations Compute Poisson brackets between arcs in the cusped lamination. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Poisson bracket Introduce cusped laminations Compute Poisson brackets between arcs in the cusped lamination. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Poisson structure Theorem The Poisson algebra of the λ -lengths of a complete cusped lamination is a Poisson cluster algebra [ Chekhov-Mazzocco. ArXiv:1509.07044] . { g s i , t j , g p r , q l } = g s i , t j g p r , q l I s i , t j , p r , q l I s i , t j , p r , q l = ϵ i − r δ s , p + ϵ j − r δ t , p + ϵ i − l δ s , q + ϵ j − l δ t , q 4 Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Decorated character variety-1 What is the character variety of a Riemann surface with cusps on its boundary? Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Decorated character variety-1 What is the character variety of a Riemann surface with cusps on its boundary? For Riemann surfaces with holes: Hom ( π 1 (Σ) , P SL 2 ( C )) / GL 2 ( C ) . Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Decorated character variety-1 What is the character variety of a Riemann surface with cusps on its boundary? For Riemann surfaces with holes: Hom ( π 1 (Σ) , P SL 2 ( C )) / GL 2 ( C ) . For Riemann surfaces with bordered cusps: Decorated character variety [ Chekhov-Mazzocco-V.R. arXiv:1511.03851] Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Decorated character variety-1 What is the character variety of a Riemann surface with cusps on its boundary? For Riemann surfaces with holes: Hom ( π 1 (Σ) , P SL 2 ( C )) / GL 2 ( C ) . For Riemann surfaces with bordered cusps: Decorated character variety [ Chekhov-Mazzocco-V.R. arXiv:1511.03851] Replace π 1 (Σ) with the groupoid G of all paths γ ij from cusp i to cusp j modulo homotopy. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Decorated character variety-1 What is the character variety of a Riemann surface with cusps on its boundary? For Riemann surfaces with holes: Hom ( π 1 (Σ) , P SL 2 ( C )) / GL 2 ( C ) . For Riemann surfaces with bordered cusps: Decorated character variety [ Chekhov-Mazzocco-V.R. arXiv:1511.03851] Replace π 1 (Σ) with the groupoid G of all paths γ ij from cusp i to cusp j modulo homotopy. Replace tr by two characters: tr and tr K . Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Cusped fat graphs Cusped fat graph (a graph with the prescribed cyclic ordering of edges entering each vertex) G g , s , n a spine of the Riemann surface Σ g , s , n with g handles, s holes and n > 0 decorated bordered cusps if (a) this graph can be embedded without self-intersections in Σ g , s , n ; (b) all vertices of G g , s , n are three-valent except exactly n one-valent vertices (endpoints of the open edges), which are placed at the corresponding bordered cusps; (c) upon cutting along all non-open edges of G g , s , n the Riemann surface Σ g , s , n splits into s polygons each containing exactly one hole and being simply connected upon contracting this hole. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Geometric laminations We call geometric cusped geodesic lamination (CGL) on a bordered cusped Riemann surface a set of nondirected curves up to a homotopy equivalence such that (a) these curves are either closed curves ( γ ) or arcs ( a ) that start and terminate at bordered cusps (which can be the same cusp); (b) these curves have no (self)intersections inside the Riemann surface (but can be incident to the same bordered cusp); (c) these curves are not empty loops or empty loops starting and terminating at the same cusp. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Decorated character variety-2 We introduce the fundamental groupoid of arcs G as the set of all directed paths γ ij : [0 , 1] → ˜ Σ g , s , n such that γ ij (0) = m i and γ ij (1) = m j modulo homotopy. The groupoid structure is dictated by the usual path–composition rules. For each m j , j = 1 , . . . , n , the isotopy group Π j = { γ jj | γ jj : [0 , 1] → ˜ Σ g , s , n , γ jj (0) = m j , γ jj (1) = m j } / { homotopy } is isomorphic to the usual fundamental group and Π j = γ − 1 ij Π i γ ij for any arc γ ij ∈ G . The decoration assigns to each arc γ ij a matrix M ij ∈ SL 2 ( R ), for example M ij = X ( k j ) LX ( z n ) R · · · LX ( z 1 ) RX ( k i ). Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Decorated character variety-3 To associate a matrix in SL 2 ( C ) - complexify the coordinates. The decorated character variety is: Hom ( G , SL 2 ( C )) / ∏ n j =1 B j , where B j is the (unipotent) Borel subgroup in SL 2 ( C ) (one Borel subgroup for each cusp) with the characters: Tr K : SL 2 ( C ) → C ( ) 0 0 M �→ Tr ( MK ) , where K = . − 1 0 Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation PVI p 1 s 3 p 3 s 1 s 2 p 2 Figure 4. The fat graph of the 4 holed Riemann sphere. The dashed geodesic is x 1 ; the solid geodesics are x 2 and x 3 . Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation Shear coordinates in the Teichm¨ uller space Fatgraph: p 1 s 3 p 3 s 1 s 2 p 2 Figure 4. The fat graph of the 4 holed Riemann sphere. The dashed geodesic is x 1 ; the solid geodesics are x 2 and x 3 . Decompose each hyperbolic element in Right, Left and Edge matrices Fock, Thurston ( ) ( ) 1 1 0 1 R := , L := , − 1 0 − 1 − 1 ( y ( ) ) 0 − exp 2 X y := . − y ( ) exp 0 2 Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation p 1 s 3 p 3 s 1 s 2 p 2 Figure 4. The fat graph of the 4 holed Riemann sphere. The dashed geodesic is x 1 ; the solid geodesics are x 2 and x 3 . The three geodesic lengths: x i = Tr ( γ jk ) p 2 2 + e − p 2 p 3 2 + e − p 3 x 1 = e s 2 + s 3 + e − s 2 − s 3 + e − s 2 + s 3 +( e 2 ) e s 3 +( e 2 ) e − s 2 p 3 2 + e − p 3 p 1 2 + e − p 1 x 2 = e s 3 + s 1 + e − s 3 − s 1 + e − s 3 + s 1 +( e 2 ) e s 1 +( e 2 ) e − s 3 p 1 2 + e − p 1 p 2 2 + e − p 2 x 3 = e s 1 + s 2 + e − s 1 − s 2 + e − s 1 + s 2 +( e 2 ) e s 2 +( e 2 ) e − s 1 Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation p 1 s 3 p 3 s 1 s 2 p 2 Figure 4. The fat graph of the 4 holed Riemann sphere. The dashed geodesic is x 1 ; the solid geodesics are x 2 and x 3 . The three geodesic lengths: x i = Tr ( γ jk ) p 2 2 + e − p 2 p 3 2 + e − p 3 x 1 = e s 2 + s 3 + e − s 2 − s 3 + e − s 2 + s 3 +( e 2 ) e s 3 +( e 2 ) e − s 2 p 3 2 + e − p 3 p 1 2 + e − p 1 x 2 = e s 3 + s 1 + e − s 3 − s 1 + e − s 3 + s 1 +( e 2 ) e s 1 +( e 2 ) e − s 3 p 1 2 + e − p 1 p 2 2 + e − p 2 x 3 = e s 1 + s 2 + e − s 1 − s 2 + e − s 1 + s 2 +( e 2 ) e s 2 +( e 2 ) e − s 1 { x 1 , x 2 } = x 1 x 2 + 2 x 3 + ω 3 , { x 2 , x 3 } = x 2 x 3 + 2 x 1 + ω 1 , { x 3 , x 1 } = x 3 x 1 + 2 x 2 + ω 2 . Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation The confluence from the cubic associated to PVI to the one associated to PV is realized by p 3 → p 3 − 2 log[ ϵ ] , in the limit ϵ → 0. We obtain the following shear coordinate description for the PV cubic: − e s 2 + s 3 + p 2 2 + p 3 2 − G 3 e s 2 + p 2 2 , x 1 = − e s 3 + s 1 + p 3 2 + p 1 2 − e s 3 − s 1 + p 3 2 − p 1 2 − G 3 e − s 1 − p 1 2 − G 1 e s 3 + p 3 2 , x 2 = − e s 1 + s 2 + p 1 2 + p 2 2 − e − s 1 − s 2 − p 1 2 − p 2 2 − e s 1 − s 2 + p 1 2 − p 2 2 − G 1 e − s 2 − p 2 2 − G x 3 = where pi 2 + e − pi p 3 G ∞ = e s 1 + s 2 + s 3 + p 1 2 + p 2 2 + p 3 2 , 2 , 2 . G i = e i = 1 , 2 , G 3 = e Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
Painlev´ e equations Geodesic lengths Decorated character variety Quantisation These coordinates satisfy the following cubic relation: x 1 x 2 x 3 + x 2 1 + x 2 2 − ( G 1 G ∞ + G 2 G 3 ) x 1 − ( G 2 G ∞ + G 1 G 3 ) x 2 − − G 3 G ∞ x 3 + G 2 ∞ + G 2 3 + G 1 G 2 G 3 G ∞ = 0 . (11) Note that the parameter p 3 is now redundant, we can eliminate it by rescaling. To obtain the correct PV- cubic, we need to pick p 3 = − p 1 − p 2 − 2 s 1 − 2 s 2 − 2 s 3 so that G ∞ = 1. Volodya Roubtsov , ITEP Moscow and LAREMA, UMR 6093 du CNRS, Universit´ Painlev´ e monodromy varieties: geometry and quantisation e d’Angers. Based on Chekhov-Mazzocco-R.
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